Break points in band structure

[chikou24i]

What do we mean by break points in band structure ? and what that sentence means : "if the band index were chosen in such a way that the energies are indexed in ascending order for any k then break points would appear in the ε_nk vs. k plot (with n fixed) wherever two lines intersect." ?
Reference: [Jenö_Sólyom,_A._Piróth]_Fundamentals_of_the_Physics of solids Volume II, page 85

 

[chikou24i]

sorry, I didn't understand what you have said.

 

[paralleltransport]

sorry, I didn't understand what you have said.

Maybe I'm wrong, but I think that it's trying to describe what happens when 2 bands intersect. If 2 bands intersect, and you index band 1 has the "lower energy" band, and band 2 as the higher energy band

$$\epsilon_1(k)$$ would look like a surface with sharp edges (discontinuous) for dirac cone type structures. Maybe the dirac cone is a break point?

https://www.google.com/search?q=dir...=OcL_Vtm9POmpjgTwlbHwDw#imgrc=FS6VEyJMufMagM: